complex base tetration program

[sheldonison]

This is the link to tetcomplex.gp pari gp code, which generates tetration for arbitrary bases in the complex plane, using the merged fixed point solution. You'll need to download par-gp.. Then load tetcomplex.gp and init(b), for a complex base, and then sexp(z) gives the resulting tetration function at tet(z). tet(0)=1.
  tetcomplex.gp (Size: 59.81 KB / Downloads: 1,408) updated March 11th, improved convergence near eta, and works with arbitrary precision settings. Previous updates improved convergence near Shell Thron boundary, and for real bases<eta, and as imag(z) gets bigger The program is now working reasonably well, but is not fully debugged. The program generalizes the ideas Mike and I talked about, and Mike implemented, for generating merged fixed point tetration solutions in the complex plane. Mike has been advocating these merged fixed point solutions as the natural extension of tetration at the real axis, for bases>eta, as well as Dimitrii Kouznetsov. This program goes beyond what Mike did, in that it works for arbitrary bases, inside and outside the Shell Thron region. For bases not too close to one and not too close to the Shell Thron boundary, this program will generate both primary fixed points, using this algorithm, and figures out if they're both repelling, or if one of them is attracting. Then it will generate both Schroder functions and their inverses for both fixed points, and use those to generate superfunctions and superfunction inverses for both fixed points. Then the program iterates, generating two theta(z) mappings, one for the upper fixed point superfunction, and one for the low fixed point superfunction. Between these, is an sexp(z) taylor series. For imaginary(z)>0.175, \( \text{sexp}(z)\approx\text{superf}(z+\theta(z)) \), and for imaginary(z)<-0.175, \( \text{sexp}(z)\approx\text{superf2}(z+\theta2(z)) \). For bases at the real axis > eta, this is mathematically identical to my Kneser tetration algorithm.

The program is fast. The program does have bugs, although it is already working surprisingly well. Also, if the base is too close to the Shell Thron boundary, the program gets really slow, and eventually stops converging. The program comes with a fast complex plane graphing algorithm, MakeGraph. So the default precision is \p 28, which allows very fast accurate results to 12 decimal digits, for making graphs. You can increase the prcecision to \p 67, and results should work for bases between 1.1 and 3. For some bases, higher precision works, up to \p 134, probably depending on the accuracy of the theta, and the superfunction and the initialization for that base. This routine lacks many of the complicated optimizations, that allow kneser.gp to work for a very wide range of real bases. The program will aid in the study of tetration in the complex plane. Below, are a few graphing examples. The program works best with imag(base)>0, and gives novel interesting solutions for some bases with imag(b)<0, where the base is inside the shell thron boundary.
- Sheldon

base=2+0.56i, which is just inside the Shell Thron boundary. This pair of solutions (this one and the one below), seems to indicate that a merged tetration solution on the Shell Thron boundary itself works!
   

base=2+0.51i, which is just outside the Shell Thron boundary, with both fixed points repelling. This is a natural extension of tetration for real bases>eta.
   

base=1.45+0.3i, which is tetration where the two superfunctions are roughly at 45 degree angles, and -135 degree angles, from the sexp(z) at z=0.
   

base=1.45-0.3i. This result totally blew me away, because the upper fixed point, for imag(z)>0, for is 1.2125955-0.6939i, which has imag(z)<0, and the lower fixed point is 1.6459+3.82111i, which has imag(z)>0. Also, the superfunction gets more chaotic as imag(z) decreases for the lower fixed point, which is also interesting. The alternative solution, rotating in the other direction around eta, is the complex conjugate solution of the solution above, which makes much more sense, with the fixed points in the "correct" locations. I wonder if this solution can be extended, as we continue rotating further around eta?
- Sheldon
   

[Kouznetsov]

1. Are the multiple cut lines unavoidable?
Can you define the superexponential with only one cut line from the point -2 for each base?
I would like to declare such superexponential as the "true and only" tetration.
2. I cite your article at http://tori.ils.uec.ac.jp/TORI/index.php...mplex_base

[sheldonison]

1. Are the multiple cut lines unavoidable?
Can you define the superexponential with only one cut line from the point -2 for each base?
I would like to declare such superexponential as the "true and only" tetration.
2. I cite your article at http://tori.ils.uec.ac.jp/TORI/index.php...mplex_base

Hey Dimitrii, the multiple cut points arises when the base is inside the Shell Thron region, because one of the two fixed points is an attracting fixed point. Such an attracting fixed point is periodic, with multiple instances of sexp(z)=0, sexp(z-1)=singularity.

If the base is outside the Shell Thron region, then both of the fixed points are repelling, and then there is only one singularity cut point in the graph, and that singularity is at z=-2. This is the case for the 2nd of the four graphs I posted, b=2+0.51i. But the first graph, case, which is nearby, with b=2+0.56i, but inside the Shell Thron boundary, has iterated singularities based on the period of the upper fixed point.
- Sheldon

[sheldonison]

Here are some plots, as the base loops around eta. The first image is for base=eta+0.25, which is tetration at the real axis. Then each subsequent image is for the base looped around eta, pi/12th of a circle, counter clockwise. Then the 12th image is for base=1.195, and then I continued around to imag(z)<0. I stopped at B=1.621-0.1768i, which didn't converge. I included it anyway, in the image, just to show what might happens. My guess is that these merged solutions cannot be continued past the 2nd Shell Thron boundary, even though they can be continued from tetration at the real axis, to the first Shell Thron boundary in the upper half of the complex plane. But continuing around to bases less than eta, and than to bases with imag(z)<0, solutions become impossible the second time you cross the Shell Thron boundary. The grid lines for all of these plots are at 5 units, with the center of the plot at z=0, where sexp(z)=1.
- Sheldon
This is the first 8 bases, looping counter clockwise around eta, starting with eta+0.25. The first four bases are on the left, and the next four bases continue on the right.
   
This is the next 8 bases, continuing counterclockwise around eta. The left four images are in the upper half of the complex plane. The right four images continue, starting at the real axis for eta-0.25, and then rotate into the lower half of the complex plane.
   
Continuing on, in the lower half of the complex plane.
   

[Kouznetsov]

Beautiful job, Sheldon!
If you put points uniformly at the circle around exp(1/e),
then with few points, you can get a good precision evaluating the derivatives of tet_b(z) with respect to b by the Cauchi contour integral. So. you can plot, for example,
d tet_b(z) / db,
d^2 tet_b(z) / db^2,
as functions of z, and make conclusion about the behavior in vicinity of b=exp(1/e).

I try to guess the result:
As b approaches exp(1/e), the quasiperiods become large; so, all the cut lines at the z plane are far away from zero (and out of the field of your pics). In this sense, in vicinity of b=exp(1/e), there should be no cutlines at all, so, b=exp(1/e) is regular point for the most of values of z. Do your evaluations allow to confirm or to refute such a guess?

Is your evaluation for fixed b and various z faster than evaluation for fixed z and varable b?

[sheldonison]

Beautiful job, Sheldon!
If you put points uniformly at the circle around exp(1/e),
then with few points, you can get a good precision evaluating the derivatives of tet_b(z) with respect to b by the Cauchi contour integral. So. you can plot, for example,
d tet_b(z) / db,
d^2 tet_b(z) / db^2,
as functions of z, and make conclusion about the behavior in vicinity of b=exp(1/e).

I try to guess the result:
As b approaches exp(1/e), the quasiperiods become large; so, all the cut lines at the z plane are far away from zero (and out of the field of your pics). In this sense, in vicinity of b=exp(1/e), there should be no cutlines at all, so, b=exp(1/e) is regular point for the most of values of z. Do your evaluations allow to confirm or to refute such a guess?

Thanks Dimitrii. Yes, my evaluations confirm such a result, but with caveat. Consider only the solutions I plotted for imag(b)>0, rotating counter clockwise, in a unit half circle with radius=0.25 around base eta=exp(1/e), and concatenate them with their imaginary conjugate solutions, rotating clockwise around base eta. This generates a full 360 degree unit circle with radius=0.25. Now, we have two different solutions for base=\( \exp(1/e)-0.25 \approx 1.19467 \), depending on whether or not the solution was generated rotating clockwise, or rotating counter clockwise. One solution is the complex conjugate of the other solution. But in fact, these two solutions are nearly identical! In fact, using the default precision for my program, \p 28, gives results accurate to only about 12 decimal digits. The imaginary component of the solution is for all intents and purposes, zero. This is somewhat analogous to the paper that you and Henryk authored, about the multiple solutions to base=sqrt(2).

So we need to use much higher precision to see what the resulting taylor series really is, so we can know how large the discontinuity is at the real axis. So we switch to \p 67, and regenerate the result, which is now accurate to nearly 32 decimal digits. Now, I display 32 decimal digits of the first 32 taylor series terms, and we can clearly see the imaginary component of the taylor series is not equal to zero, but it is small. This implies that a cauchy integral for the taylor series coefficients at this radius would be accurate to about 12 or so decimal digits, which is what I reported in our earlier email. If the radius is smaller, than the imaginary component gets arbitrarily small, as the period goes to infinity as the base approaches eta.

But, as I have shown, there are valid solutions continuing around the unit circle, past Pi in either direction! These solutions may become more and chaotic as we rotate around the unit circle. More investigation is required.

Code:

base = eta-0.25 = 1.1946678610097661336583391085964
a0=   1.0000000000000000000000000000002 - 2.5045231329042313021714830574624 E-33 *I
a1=   0.38496815270132386352241683306460 + 2.9620626097717511899678016191509 E-15*I
a2=  -0.30638546356948817486767367830749 - 2.6869898290903338543289980018355 E-12*I
a3=   0.17137553259854622103441222382774 + 4.2539626191909924099462858637538 E-12*I
a4=  -0.078842783289222721712390323634160 + 5.2707372816265845032792519650466 E-12*I
a5=   0.033397834492629539091982856016043 - 1.1834996632986949273365960313094 E-11*I
a6=  -0.014062642712262839852834627395487 - 1.0368427952696666700362295909949 E-12*I
a7=   0.0060726830996950253025451542948419 + 1.1849869471152561434791390649593 E-11*I
a8=  -0.0026850556844815823250514434924228 - 3.7602323779180141105383026598257 E-12*I
a9=   0.0012040486094043950995855470134962 - 5.3158941236931785524929546755397 E-12*I
a10= -0.00054474814804614731040426857600308 + 3.1944313543928145461027272093334 E-12*I
a11=  0.00024832392231804542064225683981716 + 1.1076037258793794680406808795498 E-12*I
a12= -0.00011400411390499149013484464217955 - 1.2224736525685550155452450286042 E-12*I
a13=  0.000052675540606501631486599712359474 - 3.1538787710070020583821885985944 E-14*I
a14= -0.000024475226952919505782311190678321 + 2.7875704395526464117303918113594 E-13*I
a15=  0.000011427590750796733085407589555046 - 4.2995304405247787558561937922391 E-14*I
a16= -0.0000053584356003719399501796824446649 - 4.1002066255892877710071670543552 E-14*I
a17=  0.0000025221448337440401171565499390582 + 1.2948108547854285868732829804724 E-14*I
a18= -0.0000011911750001530884957050568945427 + 3.7704568003050785102096244717384 E-15*I
a19=  0.00000056429134007987082478685580515917 - 2.1444364924137420108000782393165 E-15*I
a20= -0.00000026805421022158351352476965081551 - 1.4207109662028524318097129012894 E-16*I
a21=  0.00000012764981127354399472472848135343 + 2.4160910253209088209640235836972 E-16*I
a22= -0.000000060925322495377065126875790768965 - 1.6669522319047266527277839758739 E-17*I
a23=  0.000000029138684091882210545890916875488 - 1.9324014251813970993635668172572 E-17*I
a24= -0.000000013962439484810850056781055681481 + 3.6832378817727104363013010565609 E-18*I
a25=  0.0000000067020194575469197752230803984840 + 1.0388483526899569925433126202680 E-18*I
a26= -0.0000000032221401102863250739766134720131 - 3.8394830991863341982500368103067 E-19*I
a27=  0.0000000015514056726555442749481756970557 - 2.4554286735248275372944881826322 E-20*I
a28= -0.00000000074800071493876431692501570839246 + 2.6920551709726843973680186854492 E-20*I
a29=  0.00000000036110428764854932891958471091099 - 1.7058757358747842185335268010642 E-21*I
a30= -1.7453389675757007922473601336416 E-10 - 1.2990416120980836858358608407113 E-21*I
a31=  8.4451935940173048361164518224634 E-11 + 2.4640693918496310042195760772726 E-22*I

Is your evaluation for fixed b and various z faster than evaluation for fixed z and varable b?

For a given base, the evaluation of sexp(z) is nearly instantaneous once you have generated the taylor series for the sexp(z) function and the two theta(z) functions, and the two superfunctions. Results are a little slower in the complex plane, for imag(z)>0.7i or <-0.7i, since the algorithm switches to using the superfunction, and the theta(z) function needs to be evluated before the superfunction can be evaluated, and then a few iterations of exp(z) or log(z) are usually required.

It takes much much more time to regenerate these six taylor series for a different base, than it does to regenerate the solution for a different value of z, for the same base.
- Sheldon

[sheldonison]

The reason I developed the merged tetration code was to help develop some numerical evidence about the behavior to tetration in the neighborhood of eta, and whether or not eta is a branch point of complex tetration. In that sense, this thread could be viewed as a continuation of eta as a branchpoint of tetrational. From what has been posted thus far, if you consider complex tetration from etaB+r*exp(-Pi*I) ... etaB+r*exp(Pi*I), then as the radius goes to zero, the numerical results would suggest the singularity at eta is very mild, nearly vanishing as you approach eta from different directions.

Yesterday, I posted a sequence of complex plane graphs, of sexp_b(z) as the base rotates around eta. The last graph in the sequence was for base=1.621-0.177i, which is 7/8th of the way around the eta, with a 0.25 radius. This base didn't converge with the default settings. That's because my program assumes the radius of convergence for sexp(z) around z=0 is 2, but for this base, it is more like 1.2 or 1.3. To be doubly sure these results are valid, I increased the precision to 67 decimal digits, and then I had to tripled the number of terms in the sexp(x) taylor series, over what is required by default to get results accurate to 32 decimal digits. I got convergence, accurate to 32 decimal digits, and I'm plotting the results below. There's two plots below. One for base=eta+0.25*exp(1.75 pi i), and one for base=eta+0.25*exp(-0.25 pi i), corresponding to rotating 7/8th of the way around eta, counter clockwise, or rotating -1/8th of the way around eta. The two bases are the same, but the two solutions are very different.

The solution for merged tetration rotating 7/8th of the way around eta is pretty bizarre, and is near the edge of what my algorithm will converge for. A little farther, at 94.4% of the way around the circle is the Shell Thron boundary for a radius of 0.25. At we approach 94.4% of the way around the circle, to get to the Shell thron boundary, the radius of convergence for the merged solution gets arbitrarily small, and at >=90% of the way around the circle, the radius of convergence is probably<1, which breaks the tetcomplex.gp algorithm as currently written. I think the graphs will help show what is happening.

Function 1) complex graph1, 7/8th of the way around the circle
Function 2) complex graph2, -1/8th of the way around the circle

First off, contrast the difference in these two functions, with the near identical similarity with the functions +/-50% of the way around the circle. Function1-Function2, around the unit circle at z=0, differ from each other by as much as 0.03, versus a much smaller difference of <10^-11 for the two functions at eta+0.25*exp(-pi), and eta+0.25+exp(pi), going halfway around the circle clockwise and counterclockwise to eta-0.25. Second, the differences for bases<eta get arbitrarily small as the period gets arbitrarily large for bases approaching eta. That may not be the case when the base approaches the Shell Thron boundary, after rotating most of the way around the circle around eta.

Finally, this solution for 7/8th of the way is pretty bizarre. It is almost like the function is upside down. In the upper half of the complex plane, when imag(z)>0, the function converges to approximately L=1.35-1.07i, but only if real(z) is much larger than imag(z). In the lower half of the complex plane, with imag(z)<0, the function converges to L2 which is approximately 1.614+2.334i, but only when real(z) is very large negative compared to imaginary z.
- Sheldon

This is the bizarre "upside down" solution you get, rotating 7/8th of the way around eta, counter clockwise, with a radius of 0.25, based on Taylor series results generated accurate to 32 decimal digits.
   

The much more normal solution you get if you rotate -1/8th of the way around eta, with a radius of 0.25
   

[sheldonison]

I am fascinated by the concept of merged solutions on the Shell Thron boundary itself. My program is now working reasonably well, and my program will attempt numerical results fairly close to the Shell Thron boundary, but the numerical method my algorithm uses cannot work on the boundary itself, where the period for one of the fixed points is real. Initially, I would have thought that real tetration could not be extended analytically past the Shell Thron boundary, but I no longer feel this is the case due to the numerical results I'm seeing, at least for the first crossing of the Shell Thron boundary. Also, Dimitrii reports that his method works well on the Shell Thron boundary. I am also going to post my own novel results for bases on the boundary below, that were developed by looking at analytic merged tetration solutions in a region of points.

But first, I briefly want to describe the merged tetration solution and the Shell Thron boundary, and briefly describe reasons why one might not expect there to be solutions at the Shell Thron boundary boundary. Briefly, start with real tetration at the real axis, for bases>eta=e^(1/e), and rotate either clockwise, or counter clockwise. Initially, both fixed points are repelling. Consider the case where we rotate counterclockwise, through increasing imag(z). Initially, the period for the upper fixed point will have positive real and imaginary components. At the Shell Thron boundary, the upper fixed point has a real period; the lower fixed point is still repelling. I call this the first Shell Thron boundary crossing. Before continuing, I would briefly state that as we continue rotating around eta counterclockwise, we will reach the real axis for points<eta, and then continuing further, we will reach the Shell Thron boundary a second time. I believe there is a singularity at the second Shell Thron boundary crossing, but there is not a singularity at the first Shell Thron boundary crossing.

But this post is about the first Shell Thron boundary crossing. At the boundary, as long as the period is irrational (there are some additional restrictions), then the Siegel disc can be wrapped around the unit circle. And the unit circle can be unwrapped, so that the upper superfunction can be defined in the complex plane. But, the edge of the siegel disc is a fractal whose properties depend on the continued fraction of the period. So, briefly, let us briefly review the merged sexp solution generated from the two fixed points.

edit Feb 21st 2013: here, the term SuperFunction refers to the inverse abel function generated from the fixed point via the classic Schroeder function equation. \( \text{SuperFunction}(z)=\alpha^{-1}(z)=s^{-1}(\lambda^z) \), where S is the Schroeder function and \( \lambda \) is the fixed point for that Schroeder function, and \( \alpha^{-1}(z) \) is the inverse of the classic abel function generated from the Schroeder function. end edit.

if imag(z)>0 \( \text{sexp}(z)=\text{SuperFunction}_U(z+\theta_U(z)) \)
if imag(z)<0 \( \text{sexp}(z)=\text{SuperFunction}_L(z+\theta_L(z)) \)
\( \theta_U(z) \) is a 1-cyclic function which decays to a constant as z goes to +imaginary infinity, and \( \theta_L(z) \) is also a 1-cyclic function, which decays to a constant as z goes towards -imaginary infinity. At the real axis, we have the merger of the two functions, with sexp(-1)=0, and sexp(0)=1, and sexp(1)=base, and with singularities at integer values of z<=-2. theta(z) for repelling superfunctions has singularities at integer values at the real axis. theta(z) for an attracting superfunction also has singularities, but the singularities need not be at the real axis since the attracting superfunction includes the sexp(z)=0 case. This is not pertinent to my main point though. What is pertinent, is that if a merged at the Shell Thron boundary behaves analytically in a region around the z=0 point, than that implies that the \( \theta_U(z) \) function has to exactly cancel the fractal singularities of the siegel disc boundary, in the neighborhood of z=0.

This might be reason enough to consider whether such solutions exist, but that is not all! If the period is rational, then the function cannot be normalized around a siegel disc, and as far as I know, conventional methods would not be able to produce an analytic superfunction from the rationally indifferent fixed point. This means for a rational period, there is no upper superfunction, and therefore, it would seem that there cannot be a merged fixed point solution at the Shell Thron boundary. Both of these taken together should lead one to doubt whether analytic tetration can be smoothly extended through the Shell Thron boundary. The combination of these facts makes me very interested in the nature of solutions at the Shell Thron boundary.

to be continued, with graphs, and surprising numerical results ....

[sheldonison]

I am fascinated by the concept of merged solutions on the Shell Thron boundary itself. .... the numerical method my algorithm uses cannot work on the boundary itself, where the period for one of the fixed points is real.

So the algorithm, inspired by Dimitrii Kouznetsov, is to sample a bunch of different bases in the complex plane, and generate a Cauchy circle integral. Here, I sampled 52 bases, using the programs high precision mode, \p 67, and generated results accurate to ~26 decimal digits, or better, for each of those bases, for the taylor series around z=0. My goal, in picking the center, and the number of sample points, was to get the two samples closest to the Shell Thron boundary to both converge equally well, or equally poorly, and to have enough samples to get good convergance with the Cauchy circle integral. Here, the 52 samples are in magenta, equally spaced in the upper half of the complex plane. I also use the complex conjugates of those 52 points, reflected into the lower half of the complex plane. Then then 104 sample points are used to generate sexp(z) at other bases. The algorithm is to generate a Cauchy integral for each of the a_n taylor series coefficients. The algorithm generated the Cauchy integral for approximately 108 taylor series terms, which are then used to generate an extrapolated 108 taylor series polynomial sexp(z) series for some other base.
\( \text{sexp}_b(z)=\sum_{n=0}^{107}a_n\times z^n \)
   

Here, I extrapolated results for five bases.
\( b=\exp(1)\approx 2.7183 \), accurate to >25 decimal digits
\( b=\eta=\exp(1/\exp(1))\approx 1.4447 \), accurate to ~22 decimal digits
\( b=1.5 \), accurate to >25 decimal digits
\( b=\approx 1.910778+i0.33705 \), on the boundary, with "period"=5.309017.... Accurate to >25 decimal digits
\( b=\approx 1.965139+i0.441423 \), on the boundary with "period"=5. Accurate to >25 decimal digits
The accuracy was estimated by comparing B^sexp(-0.5+i) to sexp(0.5+i), for i varying from -0.85i to 0.85i.

Here is the picture for the extrapolated base with an irrationally indifferent period=5.309017.... For this base, my algorithm cannot generate results, since the base is on the Shell Thron boundary. But the extrapolation works fine. Then, from the extrapolated base, I generated a pari of exrapolated theta(z) function. This was tricky for the upper half of the complex plane, since the period is real, and it is numerically difficult to generate an accurate enough inverse Schroder and Schroder functions. I used Mike's Bell coefficient algorithm, which can generate about 40 terms, before pari-gp gets a memory failure, which is not really enough. I would need at least 200-400 hundred accurate terms to generate a theta mapping. So, I had to relax the upper theta(z) function to imaginary(z)=0.8i, due to numerical limitations. At 0.8i the extrapolated upper superfunction(z+theta(z)) and is consistent with sexp(z) to ~19 decimal digits. The extrapolated lower superfunction(z+theta(z)) is numerically straightforward, and is consistent with sexp(z) to ~27 decimal digits. I used the superfunctions, along with sexp(z) to generate this complex plane chart.
   

Here is the sexp(z) at the real axis, near z=0, where it is well behaved, with a radius of convergence=2.
   

As real z increases, the function at the real axis becomes more and more chaotic, with a smaller and smaller radius of nice behavior. Interestingly though, there are also places where graphs visually identical to the graph above can be generated, for example from z=218, to 228, because sexp(223) is nearly an even multiple of the irrational period. But, no matter how smooth they look, in these regions, sexp(z) has a poorly behaved taylor series, since in the complex plane, the function becomes very chaotic as imag(z) grows even very slightly negative. But as imag(z) grows positive in the complex plane, the function again becomes more and more well behaved, as would match the appearance of the complex plane graph.
   

So, at this point, we have numerical evidence that a Cauchy integral can be used to very accurately extrapolate sexp(z) around z=0, for arbitrary bases within a region in the complex plane. This Cauchy integral was used to extrapolate sexp(z) for a base on the Shell Thron boundary itself, with an irrationally indifferent fixed point, which would have otherwise been impossible to calculate using my algorithm. This base has a siegel disc, with a fractal at the analytic boundary. If the boundary of the siegel disc is mapped to the real axis, than this base's super function has a zero radius of converance at the real axis, which would include the points with f(z)=0, and f(z)=1, and would be a fractal at the real axis, but would be analytic for imag(z)>0. The merged fixed point mapping has cancelled out all of the fractal properties near sexp(z=0), but the fractal properties of the upper superfunction show up elsewhere at the real axis, and at the real axis the Taylor series is only well behaved near the origin. As imag(z) increases, both the upper superfunction and the sexp(z) behave similarly, and both are well behaved, and analytic. The merged sexp(z) solution approaches closer and closer to the upper fixed point as imag(z) increases.

The next post will be about the the extrapolated results to the base with a period=5, which is a rationally indifferent period, whose behavior is only nice in the region around z=0, and becomes chaotic as real(z) increases or decreases no matter what imaginary(z) is. For this base, with a rationally indifferent period=5, it can be proven that there is no upper superfunction.

For referance, these are the taylor series for B=1.9107783507843347325322538 + 0.33705042370189293106181810*I, which has an irrationally indifferent period=5.3090169943749474241022934 with a well behaved continued fraction, \( \text{period}=5+\frac{1}{\sqrt{5}+1} \), which is actually not a golden ratio base, which I originally desired, but the base still has a well behaved continued fraction.

Code:

a0=   1
a1=   0.87603021541365056116984664 + 0.13008032471692635365087803*I
a2=  -0.021250946094544245994982979 + 0.14342643007587819971477968*I
a3=   0.070834255634293596296394826 + 0.035166155746567142193324243*I
a4=  -0.017915862554963081602439323 + 0.022299417770827164459165025*I
a5=   0.0070869354742820077704754204 + 0.0033829830028820083075630988*I
a6=  -0.0044143080119538372115317067 + 0.0025005906192380288619092571*I
a7=   0.0011152454085614885854976346 - 0.0000098264079138854017724671814*I
a8=  -0.00083840864042509099295429448 + 0.00025662849558257466543978876*I
a9=   0.00024819355658581378460725956 - 0.000070987964400799443047963972*I
a10= -0.00015360831642611174382166072 + 0.000033698324233098215071290265*I
a11=  0.000057668205457701602618977422 - 0.000018550219221721496316718389*I
a12= -0.000029783983385360276430406515 + 0.0000065169753160850850752501960*I
a13=  0.000012975170098880581392903812 - 0.0000038731100268853777824141925*I
a14= -0.0000061805361348595640247094473 + 0.0000014857796611434984539602050*I
a15=  0.0000028682842307374767356663523 - 0.00000079592471107782921627179016*I
a16= -0.0000013412951061644765905252752 + 0.00000034279889982566895782662590*I
a17=  0.00000063481720197760577494545003 - 0.00000016963091873306597426448420*I
a18= -0.00000029815413315353316219123371 + 0.000000078075132168311955995811340*I
a19=  0.00000014186882793188222399540980 - 0.000000037422284926647281998857067*I
a20= -0.000000067181914621764122409978112 + 0.000000017705418362137190352782436*I
a21=  0.000000032056063988141059971868987 - 0.0000000084345842970736111406898939*I
a22= -0.000000015282025774865765776734415 + 0.0000000040298473619878370249026015*I
a23=  0.0000000073134694655411576282600721 - 0.0000000019248149223017792394054566*I
a24= -0.0000000035033495624729484052267396 + 0.00000000092336690213977526569680704*I
a25=  0.0000000016818252688465622021641994 - 0.00000000044285086139748374983229721*I
a26= -0.00000000080854518972108134244989829 + 2.1302627058705047798069927 E-10*I
a27=  0.00000000038929931586616763319838796 - 1.0253481817373489889113409 E-10*I
a28= -1.8770113710921730316120804 E-10 + 4.9445555899780883904890435 E-11*I
a29=  9.0612979769664695025880120 E-11 - 2.3868083144216509371578144 E-11*I
a30= -4.3796871344870979197019744 E-11 + 1.1536736540610877080440934 E-11*I
a31=  2.1191857722171265658652164 E-11 - 5.5822067248374999829889584 E-12*I
a32= -1.0264866586085205879636856 E-11 + 2.7038920653394572051063904 E-12*I
a33=  4.9768901912645651959864977 E-12 - 1.3109799618841163798056260 E-12*I
a34= -2.4152600334154819899587276 E-12 + 6.3620934399764172245657077 E-13*I
a35=  1.1731254377963032928386288 E-12 - 3.0901673981596221923410891 E-13*I
a36= -5.7026955416048283278774795 E-13 + 1.5021621317370808428624807 E-13*I
a37=  2.7742840881808251364064066 E-13 - 7.3078247682148758003129920 E-14*I
a38= -1.3506383749208063842248715 E-13 + 3.5577543651338624863551890 E-14*I
a39=  6.5800333416637758624409081 E-14 - 1.7332656778874696808933119 E-14*I
a40= -3.2077662162397508230456995 E-14 + 8.4496685447635291097362079 E-15*I
a41=  1.5647640545028116788131633 E-14 - 4.1217899773528856929399879 E-15*I
a42= -7.6375387630196316444244105 E-15 + 2.0118259903882388226799739 E-15*I
a43=  3.7299608417133064316781009 E-15 - 9.8251968693830224786096369 E-16*I
a44= -1.8225944961126171173779948 E-15 + 4.8009484749758473516848550 E-16*I
a45=  8.9104620237224749162074654 E-16 - 2.3471303643608856504374437 E-16*I
a46= -4.3583781614834944396572130 E-16 + 1.1480529004004690210752221 E-16*I
a47=  2.1328233587266270758186165 E-16 - 5.6181312052304511507447599 E-17*I
a48= -1.0441947695604730717827668 E-16 + 2.7505434087624103059301397 E-17*I
a49=  5.1144233632166090477248687 E-17 - 1.3472049339218024442595126 E-17*I
a50= -2.5060674484791031257939137 E-17 + 6.6013041818838107097707949 E-18*I
a51=  1.2284644356695927024778266 E-17 - 3.2359334218356336445427255 E-18*I
a52= -6.0242005986901095436856816 E-18 + 1.5868519669356280136979224 E-18*I
a53=  2.9552682184979011137118537 E-18 - 7.7845568196921167647870737 E-19*I
a54= -1.4502705147564008138508763 E-18 + 3.8201991809472432025286617 E-19*I
a55=  7.1195097988706197181770603 E-19 - 1.8753705060557620763625272 E-19*I
a56= -3.4961878451526686392994130 E-19 + 9.2094087202457755611975861 E-20*I
a57=  1.7174256072962786736719248 E-19 - 4.5239200654622455635727298 E-20*I
a58= -8.4390741097412192449934994 E-20 + 2.2229607290936158578844648 E-20*I
a59=  4.1480194938531423909435060 E-20 - 1.0926417245085846939111156 E-20*I
a60= -2.0394429169198777996597870 E-20 + 5.3721551603694317187903587 E-21*I
a61=  1.0030047086524016478434906 E-20 - 2.6420434812801584928253664 E-21*I
a62= -4.9341360539169686675496014 E-21 + 1.2997149442860420441457280 E-21*I
a63=  2.4279083789699590633684417 E-21 - 6.3954239388348474607950009 E-22*I
a64= -1.1949862795435827725550778 E-21 + 3.1477485550985733507780444 E-22*I
a65=  5.8830089138250637261708164 E-22 - 1.5496605250150056265649114 E-22*I
a66= -2.8969341756568249718903724 E-22 + 7.6308876599098472168054220 E-23*I
a67=  1.4268466517197596200202064 E-22 - 3.7584879646841266046507384 E-23*I
a68= -7.0293234206338996742023346 E-23 + 1.8516097116573383893275390 E-23*I
a69=  3.4637355383618981390675866 E-23 - 9.1239746200782705618305516 E-24*I
a70= -1.7071321708154957787456481 E-23 + 4.4968176884870150380531194 E-24*I
a71=  8.4153413406814376001667834 E-24 - 2.2166916392693332423370017 E-24*I
a72= -4.1492461807449940841829943 E-24 + 1.0929298337518721812848341 E-24*I
a73=  2.0463246549496174943395562 E-24 - 5.3908967863665571554123910 E-25*I
a74= -1.0095560829869479255852945 E-24 + 2.6602414826634015954198073 E-25*I
a75=  4.9803751930402496382264439 E-25 - 1.3123964464199691856741067 E-25*I
a76= -2.4559415252898829007647481 E-25 + 6.4619324886054124930275233 E-26*I
a77=  1.2099980575626201021595812 E-25 - 3.1754840284004279640500558 E-26*I
a78= -5.9739907765311307592761373 E-26 + 1.5676010727997093759383397 E-26*I
a79=  2.9576909721142243802271205 E-26 - 7.8018084804795765364447938 E-27*I

- Sheldon

[sheldonison]

More on the Shell Thron boundary, itself. For a rational period of 5, or any other rational period, there is no superfunction. So the function becomes more and more chaotic as real(z) increases, or decreases. For increasing z, the chaos occurs around integer values+1/2, and for decreasing so, it occurs near the negative integers, where the singularity becomes stronger, instead of decaying. There are probably additional singularities, as well. Here I graph little pieces of the merged superfunction, to show the behavior. In my previous post, the merged sexp(z) was generated from a base with an irrationally indifferent fixed point. That sexp(z) solution does not show any visible misbehavior, in similar graphs. Presumably, the misbehavior would depend on the continued fraction representation of the period. I think the misbehavior may eventually extend to arbitrarily large values of imag(z). Since there is no superfunction, my algorithm cannot extend results to increasing imag(z), but Dimitrii's method should allow extending this rationally indifferent sexp(z) function as imag(z) goes to +i infinity. Here, I graph sexp(z)-L, so that the colors loop around the fixed point in a rainbow.
   

This image shows the behavior as real(z) decreases.
   

A couple of years ago, Henryk started a thread on new results from complex dynamics. It be interesting to know what complex dynamics says about this case, where one of the fixed points is rationally indifferent, and the other is a normal repelling fixed point, and/or the case where the one of the fixed points is irrationally indifferent.
- Sheldon
For reference, these are the taylor series coefficients used for the plots in this post, which is for the merged fixpoint solution for the base whose upper superfunction has a 'period'=5, with results accurate to approximately 26 decimal digits.

Code:

B = 1.965138765983644624714311722 + 0.4412433260153932508689331862*I
a0=   1
a1=   0.9079807434839744851746261607 + 0.1591412728333623157413881350*I
a2=   0.007659492127868240456307876222 + 0.1846399330316959146545308260*I
a3=   0.06959571260339947521384981633 + 0.05494395086384639072904240777*I
a4=  -0.01853896739586610864156284686 + 0.03208183862752477800053505024*I
a5=   0.004343316287399599447846087888 + 0.006508307197205574089532061575*I
a6=  -0.005345207961363676870744658158 + 0.003585318340289147472698483962*I
a7=   0.0004398010032873102933211473086 + 0.0001619039523931708562962812843*I
a8=  -0.001003777877024937424502267473 + 0.0002862394068353525818737587700*I
a9=   0.0001479472026338298193569454915 - 0.00009953065219165022072959024958*I
a10= -0.0001662244954792588969007927661 + 0.00002235001201454921438165448298*I
a11=  0.00004655824195004755206712949921 - 0.00002767430137890164060186821929*I
a12= -0.00002876332027277035001458540190 + 0.000004231641397656991640456123169*I
a13=  0.00001183530971334214823943042012 - 0.000005341504387449057331084908997*I
a14= -0.000005612719915424328356904312019 + 0.000001289094648662858844870211635*I
a15=  0.000002702128429420899103644987746 - 0.0000009799472019399010609484192961*I
a16= -0.000001205336226991155083066773933 + 0.0000003518138615910355943883918922*I
a17=  0.0000005963874917370866392729873832 - 0.0000001924208068631061932662511999*I
a18= -0.0000002703191296883001291959068737 + 0.00000008552425286430910256950860007*I
a19=  0.0000001319738521258292228657432965 - 0.00000004105838543838365584367414149*I
a20= -0.00000006146589583669697426749838905 + 0.00000001970691268589321217243576770*I
a21=  0.00000002961711704252429566540478908 - 0.000000009210492290507201311794151223*I
a22= -0.00000001404873356874825470800388787 + 0.000000004479659273013273766147094565*I
a23=  0.000000006737179859607704090354177211 - 0.000000002109766520357551734717742807*I
a24= -0.000000003225884711771446753906410292 + 0.000000001022237304639772309102919065*I
a25=  0.000000001548114388103841631193419049 - 0.0000000004871410509379935244138690376*I
a26= -0.0000000007446997429243798877475250267 + 0.0000000002352112954763236930800349863*I
a27=  0.0000000003583501502122949980151043462 - 1.129923889224470533764393227 E-10*I
a28= -1.728603213753762812740473800 E-10 + 5.453507804323909583206114089 E-11*I
a29=  8.342050243813429497490025940 E-11 - 2.631831742274014885336360246 E-11*I
a30= -4.032925253281005910200771743 E-11 + 1.272055494611321230410440559 E-11*I
a31=  1.951147105623435138879360524 E-11 - 6.155918034173348886504797412 E-12*I
a32= -9.451578356062999050506602176 E-12 + 2.981306712035548668200384191 E-12*I
a33=  4.582421795355821824849509605 E-12 - 1.445678022921137883431517914 E-12*I
a34= -2.223852478904982341991740882 E-12 + 7.015089710943148331012188469 E-13*I
a35=  1.080154524081000844565622534 E-12 - 3.407556284021761025254119328 E-13*I
a36= -5.250739238722500796005850588 E-13 + 1.656384186376948876500405934 E-13*I
a37=  2.554423001336741090860955710 E-13 - 8.058265964053152649733211913 E-14*I
a38= -1.243596553139020331371677062 E-13 + 3.923059787337772543123214696 E-14*I
a39=  6.058562914071990590596928595 E-14 - 1.911243134624896486672673790 E-14*I
a40= -2.953544318944044594994858795 E-14 + 9.317298427938769206399126597 E-15*I
a41=  1.440754893894594649211638307 E-14 - 4.545022731062749478616983881 E-15*I
a42= -7.032251751204233617915117501 E-15 + 2.218405535233745344509350613 E-15*I
a43=  3.434356605653919441915312888 E-15 - 1.083406554742385931896859377 E-15*I
a44= -1.678151288311960576191600926 E-15 + 5.293921578047610160999371309 E-16*I
a45=  8.204295608740320516285620802 E-16 - 2.588138352126890810641316895 E-16*I
a46= -4.012970658337325798408519088 E-16 + 1.265937579047528306560052950 E-16*I
a47=  1.963794131970996574622400360 E-16 - 6.195012716053768520438385859 E-17*I
a48= -9.614408915812041114031088261 E-17 + 3.032975232054436438030567546 E-17*I
a49=  4.709098185791310260074042045 E-17 - 1.485538829966010884395285567 E-17*I
a50= -2.307458133217148892201936376 E-17 + 7.279140385851274361399816673 E-18*I
a51=  1.131106921186611173052383837 E-17 - 3.568206057029077684947420636 E-18*I
a52= -5.546774346291304158250684900 E-18 + 1.749793353533065003341975364 E-18*I
a53=  2.721059107919533582186112050 E-18 - 8.583891945510160809496403880 E-19*I
a54= -1.335334563689921674928873757 E-18 + 4.212465482148494948392331835 E-19*I
a55=  6.555278763677110023492727087 E-19 - 2.067937603451532486240278374 E-19*I
a56= -3.219110105500072144843864097 E-19 + 1.015505070445530642909563527 E-19*I
a57=  1.581317244201818746662513402 E-19 - 4.988445956813825095210712446 E-20*I
a58= -7.770265770670810750547222828 E-20 + 2.451219139803678807240155259 E-20*I
a59=  3.819283189576525330297209993 E-20 - 1.204836536477911371846248647 E-20*I
a60= -1.877814232551506691213926782 E-20 + 5.923779651999245740383199924 E-21*I
a61=  9.235151938298962165893936310 E-21 - 2.913334225847860373679672491 E-21*I
a62= -4.543098915097051302883381738 E-21 + 1.433172483793440107934695348 E-21*I
a63=  2.235493254326134998753514027 E-21 - 7.052119798843451395387359767 E-22*I
a64= -1.100281930305726854199048095 E-21 + 3.470965941697627634793739024 E-22*I
a65=  5.416772282846451016308228898 E-22 - 1.708783103833573008616087752 E-22*I
a66= -2.667348397740396154799836426 E-22 + 8.414446178301270078080072237 E-23*I
a67=  1.313767413079105214069445223 E-22 - 4.144421630153321600025135107 E-23*I
a68= -6.472241541363306324329732163 E-23 + 2.041737620118049490106781936 E-23*I
a69=  3.189228872612451515471409849 E-23 - 1.006084093575876170617783093 E-23*I
a70= -1.571839954555664790501461317 E-23 + 4.958550391145711493488891033 E-24*I
a71=  7.748421558917525267188408130 E-24 - 2.444317288792335780386575226 E-24*I
a72= -3.820431440274657698257468206 E-24 + 1.205155001518322479813597008 E-24*I
a73=  1.884134025252691322201483888 E-24 - 5.944277070332361533838162673 E-25*I
a74= -9.295301766583011369912095735 E-25 + 2.933067459410344555731401738 E-25*I
a75=  4.585498801958408021086081911 E-25 - 1.446978970199893842357274206 E-25*I
a76= -2.261579575434027128422136034 E-25 + 7.126893696589671916611800212 E-26*I
a77=  1.114376323355299788740067499 E-25 - 3.503734579154330225494917534 E-26*I
a78= -5.502806693794137845693412775 E-26 + 1.729697113477494149082120615 E-26*I
a79=  2.722297695938104032335612083 E-26 - 8.585723035060422857377695280 E-27*I
a80= -1.347403026381502750265909122 E-26 + 4.265578796441743424478689569 E-27*I
a81=  6.555286514914951073829086733 E-27 - 2.007571996524703178872730421 E-27*I